Problem: Compute the number of real solutions $(x,y,z,w)$ to the system of equations:
\begin{align*}
x &= z+w+zwx, \\ 
y &= w+x+wxy, \\
z &= x+y+xyz, \\
w &= y+z+yzw.
\end{align*}
Explanation: We can re-write the first equation as
\[x = \frac{w+z}{1-wz}.\]which is an indication to consider trigonometric substitution.

Let $x = \tan a,$ $y = \tan b,$ $z = \tan c,$ and $w = \tan d,$ where $-90^{\circ} < a,$ $b,$ $c,$ $d < 90^{\circ}$.  Then
\[\tan a = \frac{\tan d + \tan c}{1 - \tan d \tan c} = \tan (c + d).\]Similarly,
\begin{align*}
\tan b &= \tan (d + a), \\
\tan c &= \tan (a + b), \\
\tan d &= \tan (b + c).
\end{align*}Since the tangent function has period $180^\circ,$
\begin{align*}
a &\equiv c + d, \\
b &\equiv d + a, \\
c &\equiv a + b, \\
d &\equiv b + c,
\end{align*}where all the congruences are taken modulo $180^\circ.$  Adding all these congruences, we get $a + b + c + d \equiv 0.$  Then
\[a \equiv c + d \equiv -a - b,\]so $b \equiv -2a.$  Similarly, $c \equiv -2b,$ $d \equiv -2c,$ and $a \equiv -2d.$  Then
\[a \equiv -2d \equiv 4c \equiv -8b \equiv 16a,\]so $15a \equiv 0.$  Hence, $(a,b,c,d) \equiv (t,-2t,4t,-8t),$ where $15t \equiv 0.$  Since $a \equiv c + d,$
\[t \equiv 4t - 8t \equiv -4t,\]so $5t \equiv 0.$  We can check that this condition always leads to a solution, giving us $\boxed{5}$ solutions.

Note: We divided the first equation to get
\[x = \frac{w + z}{1 - wz},\]so we should check that $wz \neq 1$ for all five solutions.  If $wz = 1,$ then from the equation $x = z + w + zwx,$
\[z + w = 0.\]Then $wz = -w^2,$ which cannot be equal to 1, contradiction.  The same holds for the division in the other equations.